Abstract

The trapping and dynamic nonequilibrium spatiospectral mixing of two nonresonant picosecond pulses injected into the active area of a broad-area semiconductor laser are investigated on the basis of spatiotemporal numerical simulations. The spatiotemporal dynamics of the nonequilibrium Wigner distributions of the charge carriers and the interband polarization reveal the spatiospectral wave mixing and spatiospectral nature of self-focusing, propagation-induced hole burning, dynamic nonlinear waveguiding, and interference effects. This interplay of effects is identified as the origin of the pulse merging and of the subsequent formation of a trapped pulse that is reminiscent of a spatial optical soliton.

© 1998 Optical Society of America

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  1. A. Schulze, A. Knorr, and S. W. Koch, “Pulse propagation and many-body effects in semiconductor four-wave mixing,” Phys. Rev. B 51, 10601–10609 (1995).
    [CrossRef]
  2. S. Hughes, A. Knorr, and S. W. Koch, “Interplay of optical dephasing and pulse propagation in semiconductors,” J. Opt. Soc. Am. B 14, 754–760 (1997).
    [CrossRef]
  3. S. Hughes, “Polarization decay in ultrafast collinear nondegenerate four-wave mixing in a semiconductor amplifier,” Opt. Lett. 23, 948–950 (1998).
    [CrossRef]
  4. O. Hess, S. W. Koch, and J. V. Moloney, “Filamentation and beam propagation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 31, 35–43 (1995).
    [CrossRef]
  5. O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor laser. II. Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996).
    [CrossRef] [PubMed]
  6. I. Fischer, O. Hess, W. Elsässer, and E. Göbel, “Complex spatio-temporal dynamics in the nearfield of a broad-area semiconductor laser,” Europhys. Lett. 35, 579–584 (1996).
    [CrossRef]
  7. M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 33, 3342–3350 (1988).
    [CrossRef]
  8. H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
    [CrossRef] [PubMed]
  9. O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor lasers. I. Theoretical description,” Phys. Rev. A 54, 3347–3359 (1996).
    [CrossRef] [PubMed]
  10. E. Gehrig and O. Hess, “Nonequilibrium spatio-temporal dynamics of the Wigner-distributions in broad-area semiconductor lasers,” Phys. Rev. A 57, 2150–2163 (1998).
    [CrossRef]
  11. O. Hess and T. Kuhn, “Spatio-temporal dynamics of semiconductor lasers: theory, modeling and analysis,” Prog. Quantum Electron. 20, 85–179 (1996).
    [CrossRef]
  12. H. Haug, “Microscopic theory of the optical band edge nonlinearities,” in Optical Nonlinearities and Instabilities in Semiconductors, H. Haug, ed. (Academic, New York, 1988), pp. 53–81.
  13. W. W. Chow, S. W. Koch, and M. S. III, Semiconductor-Laser Physics (Springer-Verlag, Berlin, 1994).
  14. O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).
  15. G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).
  16. R. Zimmermann, “Nonlinear optics and the Mott transition in semiconductors,” Phys. Status Solidi B 146, 371–384 (1988).
    [CrossRef]
  17. Although the self-focusing processes may in general depend on the particular spatial shape of the input pulses and their power densities, the conditions assumed here can be considered typical; i.e., within a large variation of input intensities, only minor quantitative changes in the behavior have been observed in the simulations.

1998

S. Hughes, “Polarization decay in ultrafast collinear nondegenerate four-wave mixing in a semiconductor amplifier,” Opt. Lett. 23, 948–950 (1998).
[CrossRef]

E. Gehrig and O. Hess, “Nonequilibrium spatio-temporal dynamics of the Wigner-distributions in broad-area semiconductor lasers,” Phys. Rev. A 57, 2150–2163 (1998).
[CrossRef]

1997

1996

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor laser. II. Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996).
[CrossRef] [PubMed]

I. Fischer, O. Hess, W. Elsässer, and E. Göbel, “Complex spatio-temporal dynamics in the nearfield of a broad-area semiconductor laser,” Europhys. Lett. 35, 579–584 (1996).
[CrossRef]

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor lasers. I. Theoretical description,” Phys. Rev. A 54, 3347–3359 (1996).
[CrossRef] [PubMed]

O. Hess and T. Kuhn, “Spatio-temporal dynamics of semiconductor lasers: theory, modeling and analysis,” Prog. Quantum Electron. 20, 85–179 (1996).
[CrossRef]

1995

A. Schulze, A. Knorr, and S. W. Koch, “Pulse propagation and many-body effects in semiconductor four-wave mixing,” Phys. Rev. B 51, 10601–10609 (1995).
[CrossRef]

O. Hess, S. W. Koch, and J. V. Moloney, “Filamentation and beam propagation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 31, 35–43 (1995).
[CrossRef]

1989

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

1988

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 33, 3342–3350 (1988).
[CrossRef]

R. Zimmermann, “Nonlinear optics and the Mott transition in semiconductors,” Phys. Status Solidi B 146, 371–384 (1988).
[CrossRef]

Elsässer, W.

I. Fischer, O. Hess, W. Elsässer, and E. Göbel, “Complex spatio-temporal dynamics in the nearfield of a broad-area semiconductor laser,” Europhys. Lett. 35, 579–584 (1996).
[CrossRef]

Fischer, I.

I. Fischer, O. Hess, W. Elsässer, and E. Göbel, “Complex spatio-temporal dynamics in the nearfield of a broad-area semiconductor laser,” Europhys. Lett. 35, 579–584 (1996).
[CrossRef]

Gehrig, E.

E. Gehrig and O. Hess, “Nonequilibrium spatio-temporal dynamics of the Wigner-distributions in broad-area semiconductor lasers,” Phys. Rev. A 57, 2150–2163 (1998).
[CrossRef]

Göbel, E.

I. Fischer, O. Hess, W. Elsässer, and E. Göbel, “Complex spatio-temporal dynamics in the nearfield of a broad-area semiconductor laser,” Europhys. Lett. 35, 579–584 (1996).
[CrossRef]

Haug, H.

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

Hess, O.

E. Gehrig and O. Hess, “Nonequilibrium spatio-temporal dynamics of the Wigner-distributions in broad-area semiconductor lasers,” Phys. Rev. A 57, 2150–2163 (1998).
[CrossRef]

O. Hess and T. Kuhn, “Spatio-temporal dynamics of semiconductor lasers: theory, modeling and analysis,” Prog. Quantum Electron. 20, 85–179 (1996).
[CrossRef]

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor lasers. I. Theoretical description,” Phys. Rev. A 54, 3347–3359 (1996).
[CrossRef] [PubMed]

I. Fischer, O. Hess, W. Elsässer, and E. Göbel, “Complex spatio-temporal dynamics in the nearfield of a broad-area semiconductor laser,” Europhys. Lett. 35, 579–584 (1996).
[CrossRef]

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor laser. II. Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996).
[CrossRef] [PubMed]

O. Hess, S. W. Koch, and J. V. Moloney, “Filamentation and beam propagation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 31, 35–43 (1995).
[CrossRef]

Hughes, S.

Knorr, A.

S. Hughes, A. Knorr, and S. W. Koch, “Interplay of optical dephasing and pulse propagation in semiconductors,” J. Opt. Soc. Am. B 14, 754–760 (1997).
[CrossRef]

A. Schulze, A. Knorr, and S. W. Koch, “Pulse propagation and many-body effects in semiconductor four-wave mixing,” Phys. Rev. B 51, 10601–10609 (1995).
[CrossRef]

Koch, S. W.

S. Hughes, A. Knorr, and S. W. Koch, “Interplay of optical dephasing and pulse propagation in semiconductors,” J. Opt. Soc. Am. B 14, 754–760 (1997).
[CrossRef]

A. Schulze, A. Knorr, and S. W. Koch, “Pulse propagation and many-body effects in semiconductor four-wave mixing,” Phys. Rev. B 51, 10601–10609 (1995).
[CrossRef]

O. Hess, S. W. Koch, and J. V. Moloney, “Filamentation and beam propagation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 31, 35–43 (1995).
[CrossRef]

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 33, 3342–3350 (1988).
[CrossRef]

Kuhn, T.

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor lasers. I. Theoretical description,” Phys. Rev. A 54, 3347–3359 (1996).
[CrossRef] [PubMed]

O. Hess and T. Kuhn, “Spatio-temporal dynamics of semiconductor lasers: theory, modeling and analysis,” Prog. Quantum Electron. 20, 85–179 (1996).
[CrossRef]

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor laser. II. Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996).
[CrossRef] [PubMed]

Lindberg, M.

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 33, 3342–3350 (1988).
[CrossRef]

Moloney, J. V.

O. Hess, S. W. Koch, and J. V. Moloney, “Filamentation and beam propagation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 31, 35–43 (1995).
[CrossRef]

Schulze, A.

A. Schulze, A. Knorr, and S. W. Koch, “Pulse propagation and many-body effects in semiconductor four-wave mixing,” Phys. Rev. B 51, 10601–10609 (1995).
[CrossRef]

Zimmermann, R.

R. Zimmermann, “Nonlinear optics and the Mott transition in semiconductors,” Phys. Status Solidi B 146, 371–384 (1988).
[CrossRef]

Europhys. Lett.

I. Fischer, O. Hess, W. Elsässer, and E. Göbel, “Complex spatio-temporal dynamics in the nearfield of a broad-area semiconductor laser,” Europhys. Lett. 35, 579–584 (1996).
[CrossRef]

IEEE J. Quantum Electron.

O. Hess, S. W. Koch, and J. V. Moloney, “Filamentation and beam propagation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 31, 35–43 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. A

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor laser. II. Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996).
[CrossRef] [PubMed]

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

O. Hess and T. Kuhn, “Maxwell–Bloch equations for spatially inhomogeneous semiconductor lasers. I. Theoretical description,” Phys. Rev. A 54, 3347–3359 (1996).
[CrossRef] [PubMed]

E. Gehrig and O. Hess, “Nonequilibrium spatio-temporal dynamics of the Wigner-distributions in broad-area semiconductor lasers,” Phys. Rev. A 57, 2150–2163 (1998).
[CrossRef]

Phys. Rev. B

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 33, 3342–3350 (1988).
[CrossRef]

A. Schulze, A. Knorr, and S. W. Koch, “Pulse propagation and many-body effects in semiconductor four-wave mixing,” Phys. Rev. B 51, 10601–10609 (1995).
[CrossRef]

Phys. Status Solidi B

R. Zimmermann, “Nonlinear optics and the Mott transition in semiconductors,” Phys. Status Solidi B 146, 371–384 (1988).
[CrossRef]

Prog. Quantum Electron.

O. Hess and T. Kuhn, “Spatio-temporal dynamics of semiconductor lasers: theory, modeling and analysis,” Prog. Quantum Electron. 20, 85–179 (1996).
[CrossRef]

Other

H. Haug, “Microscopic theory of the optical band edge nonlinearities,” in Optical Nonlinearities and Instabilities in Semiconductors, H. Haug, ed. (Academic, New York, 1988), pp. 53–81.

W. W. Chow, S. W. Koch, and M. S. III, Semiconductor-Laser Physics (Springer-Verlag, Berlin, 1994).

O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).

G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).

Although the self-focusing processes may in general depend on the particular spatial shape of the input pulses and their power densities, the conditions assumed here can be considered typical; i.e., within a large variation of input intensities, only minor quantitative changes in the behavior have been observed in the simulations.

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Figures (2)

Fig. 1
Fig. 1

Snapshots of the intensity distribution in the active area of a broad-area amplifier for two propagating pulses of length 2 ps and wavelengths λ 1 = 755   nm (injected from the upper left) and λ 2 = 760   nm (injected from the lower left). The time between the images is 2.5 ps.

Fig. 2
Fig. 2

Snapshots of the nonequilibrium Wigner distributions of electrons δ f e (left), the real part p nl (center), and the imaginary part p nl (right) of the interband polarization at (a)–(c) t 1 , (d)–(f) t 2 = t 1 + 2.5   ps , and (g)–(i) t 3 = t 1 + 5   ps .

Tables (1)

Tables Icon

Table 1 Fundamental Material and Device Parameters of the Broad-Area (GaAs–AlGaAs) Semiconductor Lasers

Equations (20)

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± z   E ± ( r ,   t ) + n l c t   E ± ( r ,   t )
= i 2 1 K z 2 x 2   E ± ( r ,   t ) - α 2 + i η E ± ( r ,   t ) + i 2 Γ n l 2 0 L   P nl ± ( r ,   t ) ,
E + ( x ,   z = 0 ,   t ) = - R 1 E - ( x ,   z = 0 ,   t ) + E 1 ( x ,   z = 0 ,   t ) + E 2 ( x ,   z = 0 ,   t ) ,
E - ( x ,   z = L ,   t ) = - R 2 E - ( x ,   z = L ,   t ) .
E 1 , 2 ( x ,   z = 0 ,   L ,   t ) = E 1 , 2 0 ( x ,   z = 0 ,   t ) T 1 exp ( - i ω 1 , 2 t ) × exp - ( x - x 1 , 2 0 ) 2 w × exp 2 π i   ( x - x 1 , 2 0 ) λ 1 , 2 sin ( α 1 , 2 ) .
P nl ± ( r ,   t ) = 2 V - 1 k d cv p nl ± ( k ,   r ,   t ) ,
t   f e , h ( k ,   r ,   t ) = g ( k ,   r ,   t ) - γ e , h ( k ,   N ) × [ f e , h ( k ,   r ,   t ) - f eq e , h ( k ,   r ,   t ) ] + Λ e , h ( k ,   r ,   t ) - Γ sp ( k ,   T l ) f e ( k ,   r ,   t ) f h ( k ,   r ,   t ) - γ nr f e , h ( k ,   r ,   t ) ,
t   p nl ± ( k ,   r ,   t ) = - [ i ω ¯ ( k ,   T l ) + γ p ( k ,   N ) ] p nl ± ( k ,   r ,   t ) + 1 i   d cv E ± ( r ,   t ) [ f e ( k ,   r ,   t ) + f h ( k ,   r ,   t ) ] ,
g ( k ,   r ,   t ) = - 1 4   d cv Im [ E + ( r ,   t ) p nl + * ( k ,   r ,   t ) + E - ( r ,   t ) p nl - * ( k ,   r ,   t ) ] .
Λ e , h ( k ,   r ,   t ) = Λ ( r ,   t ) f eq e , h ( k ,   r ,   t ) [ 1 - f e , h ( k ,   r ,   t ) ] { V - 1 k f eq e , h ( k ,   r ,   t ) [ 1 - f e , h ( k ,   r ,   t ) ] }
t   N ( r ,   t ) = [ D f N ( r ,   t ) ] + Λ ( r ,   t ) + G ( r ,   t ) - γ nr N ( r ,   t ) - W ( r ,   t ) ,
G ( r ,   t ) = χ 0 2   [ | E + ( r ,   t ) | 2 + | E - ( r ,   t ) | 2 ] - 1 4 Im [ E + ( r ,   t ) P nl + * ( r ,   t ) + E - ( r ,   t ) P nl - * ( r ,   t ) ] ,
W ( r ,   t ) = V - 1 k Γ sp ( k ,   T l ) f e ( k ,   r ,   t ) f h ( k ,   r ,   t ) ,
ω ¯ ( k ,   T l ) = E g ( T l ) + k 2 2 m r + δ E ( r ,   t ,   T l ) - ω = ω T ( k ,   T l ) - ω
Γ sp ( k ,   T l ) = n l 0 π   | d cv | 2 E g ( T l ) + 2 k 2 2 m r 3
δ E ( r ,   t ,   T l ) = E 0 - a [ N ( r ,   t ) a 0 3 E 0 2 ] 1 / 4 [ N ( r ,   t ) a 0 3 E 0 2 + b 2 ( k b T l ) 2 ] 1 / 4 ,
T pl e , h ( r ,   t ) t = J u e , h ( r ,   t )   u e , h ( r ,   t ) t - J N e , h ( r ,   t )   N ( r ,   t ) t .
J u e , h ( r ,   t ) = N μ e , h u e , h ( r ,   t ) T pl e , h ( r ,   t ) N ( r ,   t ) μ e , h ( r ,   t ) - u e , h ( r ,   t ) μ e , h ( r ,   t ) N ( r ,   t ) T pl e , h ( r ,   t ) - 1 ,
J N e , h ( r ,   t ) = u e , h μ e , h u e , h ( r ,   t ) T pl e , h ( r ,   t ) N ( r ,   t ) μ e , h ( r ,   t ) - u e , h ( r ,   t ) μ e , h ( r ,   t ) N ( r ,   t ) T pl e , h ( r ,   t ) - 1 ,
T ˙ l ( r ,   t ) = - γ a [ T l ( r ,   t ) - T a ] + ph 1 τ ph e × [ T pl e ( r ,   t ) - T l ( r ,   t ) ] + J u e ( r ,   t )   1 π 2 1 V k 2 k 2 2 m e 1 τ po e × [ f e ( k ,   r ,   t ) - f eq e ( k ,   r ,   t ) ] + ph 1 τ ph h   [ T pl h ( r ,   t ) - T l ( r ,   t ) ] + J u h ( r ,   t )   1 π 2 1 V k 2 k 2 2 m h 1 τ po h   [ f h ( k ,   r ,   t ) - f eq h ( k ,   r ,   t ) ] + ω γ nr N ( r ,   t ) + J   2 R c q A 2 V BAL .

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